Optimizing the Optical Calibration Performance of a Multi-Object Adaptive Optics Instrument

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by

Laurie Nhu An Pham

M.Sc., Ecole Nationale Sup´erieur d’Ing´enieurs de Caen, France, 2007

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF APPLIED SCIENCE

in the Department of Mechanical Engineering

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Optimizing the Optical Calibration Performance of a Multi-Object Adaptive Optics Instrument

by

Laurie Nhu An Pham

M.Sc., Ecole Nationale Sup´erieur d’Ing´enieurs de Caen, France, 2007

Supervisory Committee

Dr. Colin Bradley, Supervisor

(Department of Mechanical Engineering)

Dr. Carlos Correia, Departmental Member (Department of Mechanical Engineering)

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Supervisory Committee

Dr. Colin Bradley, Supervisor

(Department of Mechanical Engineering)

Dr. Carlos Correia, Departmental Member (Department of Mechanical Engineering)

ABSTRACT

Multi-Object Adaptive Optics (MOAO) is an adaptive optics technique being developed for Extremely Large Telescopes that will allow simultaneous observation of approximately 20 targets in a several arc-minute field of regard. Raven is an MOAO pathfinder developed by the Adaptive Optics Laboratory of the University of Victoria, in collaboration with the National Research Council of Canada and the Subaru Telescope. It will be the first MOAO instrument on a 8-m class telescope, will demonstrate that MOAO technical challenges such as open-loop control and calibration are achievable on-sky and will deliver science results using three natural guide stars and two science arms on ∼ 3.50 field-of-regard. The open-loop approach makes the need for calibration even more crucial.

An important part of the calibration process resides in the misregistration of the wavefront sensors (WFSs) with the deformable mirrors (DMs) because the sensing elements are located before the correcting ones. This problem is solved using a cal-ibration DM seen by all WFSs in the system that permits the open-loop WFS to be registered to the science DMs. The method developed in this thesis registers the position of the DM actuators to the WFSs and gives misregistration values. These results are then used to better align the instrument, to have a better knowledge of the positions of the different optical components and generate new ways to perform the AO correction. Using the registration parameters results, synthetic interaction matrices are created in order to improve the AO correction. Calibration tests are also presented in this thesis. They show complementary tests to the expected requirements to expand the knowledge of the calibration unit behaviour.

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List of Tables

Table 5.1 Values read on the photodiode placed in front of the CU output. 46 Table 5.2 Exposure times for each filter. . . 47 Table 5.3 Density filters test results. . . 47 Table 5.4 Registration parameters between CDM and all WFS. The scale

unit is subaperture per actuator pitch. . . 52 Table 5.5 Registration parameters between CDM and CLWFS. The scale

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List of Figures

Figure 1.1 Illustration of anisoplanatism. The turbulence in the cylinder associated with a distant object (yellow) is not the same as the one associated with a nearby distant object (blue). . . 2 Figure 1.2 Calibration unit system. . . 4 Figure 1.3 Functional optical block diagram of RAVEN. Dashed blocks are

deployable. Raven consists of 8 main subsystems: the deploy-able Calibration Unit, the Open-Loop NGSWFSs, the Science Pick-offs, the Science Relays, the Closed-Loop NGS Truth/Fig-ure WFSs, the Beam Combiner, the LGSWFS and the Acquisi-tion Camera. . . 6 Figure 1.4 Top view of the Raven computer-aided design. . . 7 Figure 1.5 Top view of the Raven bench. . . 7 Figure 1.6 Field dependent aberrations in the field of view of a telescope.

When changing the pick-off position in the focal plane (left), the field dependent aberration seen depends on that position (right). 9 Figure 1.7 Simplified diagram illustrating how the interaction matrices are

record for the calibration. The light beam is generated in the calibration unit by the natural guide star source. It is bounces off the calibration DM (CDM). Then 2 pick-offs send the light to the open-loop wavefront sensor (OLWFS) and the science path with a science DM and a closed-loop WFS. . . 10 Figure 1.8 Illustration of the science deformable mirror actuators as seen

by the OLWFS. The three WFSs are rotated with respect to the science DM by −90◦, 180◦ and 90◦ . . . 15 Figure 2.1 Illustration of the misregistration parameters. The black grid is

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Figure 3.1 Experimental interaction matrix between a CLWFS and a SDM. Each column of the matrix represents the coordinates of the WFS

slopes during an actuator poke. . . 20

Figure 3.2 Example of WFS reference positions (red) and slopes (blue) when actuator 107 is poked. . . 20

Figure 3.3 Illustration of how the intersection of the slopes gives the ac-tuator position (blue lines). For small spot displacements, the accuracy is lost and intersect the other lines far from the actua-tors position (red line). . . 21

Figure 3.4 Illustration of the optimization function for two slopes. The cross represents the point to optimize. . . 26

Figure 3.5 The actuators outside the WFS pupil are hard to locate so they are filtered. . . 27

Figure 3.6 The actuator mapping is applied to DM with a different number of actuators: 145 for SDM and 277 for CDM. . . 28

Figure 3.7 Comparison of 2 actuator maps. The red one is the reference. The blue one is rotated by an angle of 40◦ and shifted by 5 percent of a subaperture on the WFS. The blue lines show the corresponding actuators. . . 29

Figure 3.8 Illustration of the combination of distances. The blue dots rep-resent the actuators positions and the arrows reprep-resent the dis-tances between all the possible combinations of these points. . . 29

Figure 3.9 Experimental results of the extraction of the registration param-eters from a SDM-to-CLWFS interaction matrix. . . 30

Figure 4.1 Wavefront measurement setup with a Zygo interferometer (left) and a DM (right). . . 33

Figure 4.2 (a) Zygo CDM phase and (b) influence functions. . . 34

Figure 4.3 (a) Synthetic CDM phase and (b) influence functions. . . 35

Figure 4.4 (a) Experimental and (b) synthetic SDM2-to-CLWFS2 interac-tion matrices. . . 39

Figure 4.5 (a) Experimental interaction and (b) command matrices. . . 40

Figure 4.6 (a) Experimental interaction and (b) command matrices. . . 41

Figure 4.7 Phase fitting. . . 42

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Figure 5.2 Stability test with DM commands sent to get the best flat. Mea-surements taken during 16 hours. . . 45 Figure 5.3 Repeatability measurements in function of the time for the 11

phase screens. The tip/tilt is removed so that only the DM error would appear. . . 46 Figure 5.4 Photodiode power in function of the source power for the first 3

filters. . . 47 Figure 5.5 Images from the NGS source through each filter taken with an

Andor camera. The exposure times have been chosen to use the full dynamic range of the camera. The images are defocused to use more pixels for a better averaged result and to avoid having a focused point on potential dead pixels. . . 48 Figure 5.6 GlobalMask : Map of useful subapertures. A white cell

corre-sponds to 1, a black cell to 0. . . 50 Figure 5.7 Illustration of the 3 masks isolating the zones in the WFS frame

for each poked actuator. A white cell corresponds to 1, a black cell to 0. . . 51 Figure 5.8 WFS display with 3 actuators (yellow crosses) for alignment.

The red cross is the center of the WFS. The yellow circle is the deducted center of the DM found using the 3 other actuator measurements. . . 51 Figure 5.9 The Zernike coefficients are presented on maps of pinhole

posi-tions from the calibration unit. . . 53 Figure 5.10Synthetic Interaction Matrices. . . 56 Figure 5.11Open-Loop correction of a ground layer turbulence generated

by the CDM. The correction is performed using the data from OLWFS2 and corrected on SDM1 with the residue measured by CLWFS1 . . . 58 Figure 5.12Comparison of experimental and synthetic command matrix

per-formances. . . 60 Figure B.1 Experimental Interaction Matrices. . . 83

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ACKNOWLEDGEMENTS I would like to thank:

Dr. Colin Bradley for allowing me to participate in such a big and exciting project. I am always amazed at how an instrument like Raven gets created, funded and built.

Dr. David Andersen for taking the time to help me with my thesis. I greatly appreciated the support.

Dr. Carlos Correia for being part of my committee and for your interest in my work.

Dr. Olivier Lardi`ere for being able to explain to me so many hard-to-grasp con-cepts with ease. Your sense of humour made the gruelling hours of Raven testing in the laboratory really fun.

Dr. C´elia Blain for being so helpful in the Canadian labyrinth, especially the UVic obstacle course. Thank you for listening to my rants and funny stories, and also for sharing so many nap-inducing meals.

Lori Muck for making the administrative part of Raven really easy.

The Raven Team for being such a lovely group of people. Working in a fun and helpful environment like this one is something I value.

I would also like to thank all the people I met while in Canada. Life is not only about work and I am glad to have had so many new experiences and to have discovered so many cultures and countries. I would like to thank all the friends I made from all over the world. Thank you to all the exchangers from 2011 for kickstarting my first year in Canada with an explosion of fun that I still remember very fondly up to this day. You set the bar so high that I knew it could not be repeated the following years. A big thank you to my duck-duck-duck food fanatics Mariam ”I’ll come if there’s food involved” Ghani and Camille ”we need more food” Hoang. Only you could make Canada a place where I eat more duck than when in France. I cannot thank you enough for all the food memories I have and all the cooking tricks you taught me. Have I said ”duck” yet ?

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Speaking of duck, I also have to thank the best roommate ever: Ashley ”Sour Wolf” Field. ”I volunteer Sophie as tribute”, ”That’s Woody Harrelson !”, ”Ephemere” (Angie Pass, never stop being so funny !) and ”you look so peaceful when you sleep” are some of the few quotes that always put a smile on my face. Thank you Krista ”Skinny Girl” Conle for all the movies we watched together with Mike ”Let’s climb a tree” Irvine. But no thanks for the awful ”The Covenant” and ”Priest”. Thanks Aidan for being ”phat” with me and for all the sweat when Monica’ing. Thanks K´evin for your endless amount of silliness.

And to my friends Tingting, Fanfan and Aymeric that I left in France when starting this big Canadian adventure, a big thank you for just being the same people I always knew every time I come back to visit. I cannot wait to play with your little aliens and predators.

Finally, I’d like to thank ”Observatoire de Paris” family. You guys made me want to expand my horizons beyond what I had and it led me to incredible experiences. An extra special thank you to my Persee-Cutters, Julien ”10 minutes `a perdre” Lozi, Em-ilie ”Kiwi” Lhom´e and Sophie ”Citron” Jacquinod, for making work my abs everytime I talk to you (I sound so serious here).

There are so many friends that I’m forgetting so if your name starts with a letter between A and Z, thank you !

And last but not least, my Phamily who supports me in everything I do and who believes in me more than I do. I just have the best parents in the world.

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DEDICATION

This is dedicated to my best cheerleaders and family: Papa, Maman and Phiphi.

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Introduction

1.1 Multi-Object Adaptive Optics (MOAO)

The field of view (FoV) of classical adaptive optics (AO) systems is limited by an effect known as anisoplanatism [1]. For a given telescope pointing, the light from a distant object is perturbed by the turbulence in a cylinder of the atmosphere (with a diameter equal to the telescope primary mirror). Light from a nearby distant object will pass through an overlapping, but non-identical cylinder of turbulence on its way to the telescope (figure 1.1). In classical AO systems, one wavefront sensor (WFS) will pick-off light from a single, relatively bright point source, and a deformable mirror (DM) will be commanded to correct the incoming wavefront to null out the wavefront error induced by the turbulence along a single line-of-sight (within a single cylinder). The AO correction for a nearby object will be inferior because it will be viewed through a slightly different cylinder of turbulence. The isoplanatic angle can be described as the angular distance from the guide star at which the wavefront error is below 1 radian, and it is typically about 20 arcseconds.

To enlarge the isoplanatic angle, one can place multiple DMs in series, each con-jugated to a different atmospheric altitude. This Multi-Conjugate AO (MCAO) [2, 3] approach can be used to enlarge the FoV to an arcminute or two, but the performance will ultimately still be limited by generalized anisoplanatism. The FoV can be further enlarged by adding more DMs in series, to remove the turbulence generated at even more atmospheric heights. However, the complexity of the MCAO system rises (and the throughput falls) with each additional DM relay.

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Figure 1.1: Illustration of anisoplanatism. The turbulence in the cylinder associated with a distant object (yellow) is not the same as the one associated with a nearby distant object (blue).

AO corrections can be applied between 5 and 10 arcminutes. An MOAO system selec-tively corrects for turbulence in the localised region around multiple science targets; i.e. a multiplex advantage is realized. Therefore, if the turbulent volume over the whole telescope FOV can be sensed and modelled (tomographically), then a probe with an embedded DM can be positioned anywhere in the field of regard. This en-ables optimal turbulence correction at each target’s location. Measurement of the turbulent volume utilizes wavefront data from multiple sensors that probe different lines-of-sight through the atmosphere to their respective guide stars. Once the in-formation from these multiple WFSs is combined into a single tomographic model of the turbulence, it is straight-forward to imagine multiple science pick-offs, each incorporating its own DM, feeding multiple integral field spectrographs [7, 8].

MOAO systems are planned for integration with the next generation of extremely large, ground-based optical telescopes (ELTs) . Their large FOV will enable roughly 20 science targets to be imaged and corrected using MOAO [4]. However, the WFSs will be common to all science channels and, therefore, placed upstream of each DM in an open-loop architecture. Such MOAO systems will have a field of regard of a few arcminutes to observe 20 science targets at the same time. The planned instruments for the two projects of ELTs are: IRMOS [9] for the Thirty Meter Telescope [10] [11] and EAGLE [12] for the European Extremely Large Telescope [13].

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1.2 The Raven Multi-Object Adaptive Optics

Demon-strator

Raven [14] will be the first multi-object adaptive optics instrument on an 8-m class telescope feeding an AO-optimized science instrument, the Subaru InfraRed Camera and Spectrograph (IRCS) [15]. A functional block diagram of the system is shown in figure 1.3 and a top view of the computer-aided design is shown in figure 1.4. The view of the Raven bench is shown in figure 1.5.

Raven consists of 8 main optical subsystems:

• A deployable Calibration Unit (CU) [16] which functions as both a telescope simulator and a turbulence generator. It contains an array of off-axis natural guide star (NGS) sources and one on-axis Laser Guide Star (LGS) source. Light from the CU feeds the three open-loop (OL) wavefront sensors, the LGSWFS and two science arms. The three functions of the CU are: 1) to help align other Raven subsystems, 2) to calibrate the AO system (generate interaction matrices and measure field-dependent non-common path aberrations), and 3) to test the MOAO correction with turbulence generated by two rotating phase screens and a ground-conjugated DM (Figure 1.2).

• Three NGS OL Shack-Hartmann WFSs that are mounted on x-y transla-tion stages to prevent the pupil from rotating on the WFS lenslet array with respect to the DMs.

• An on-axis LGSWFS which will be fed by the Subaru Sodium beacon in order to improve AO correction and sky coverage.

• Two science pick-off arms consisting of a mirror mounted on an r −θ arm followed by a trombone mirror that keeps the optical path length constant. • A science relay for each arm containing a DM which is a custom ALPAO DM

with 11x11 actuators over a 25 mm aperture.

• A figure source and closed-loop (CL) WFS that are employed to measure the DM shape (using the figure source); calibrate the bench or evaluate MOAO performance; or, allow classical AO system operation.

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• A beam combiner also contains a K-mirror so that extended objects can be properly aligned onto the slit. After the science relay, light from both arms of the system are combined so that the common beam shares an identical exit pupil and provides two adjacent 4 arcsecond science fields to the single IRCS slit.

• An acquisition camera that can be used to determine the telescope pointing and ensure that shadows of the probe arms fall over the NGSs and science targets.

Figure 1.2: Calibration unit system.

The science gain achievable by Raven, in comparison to classical AO systems such as Subaru’s AO188 [17], will be modest because Raven only has two science channels. Nevertheless, the 8 m aperture of the Subaru Telescope enables science that is not achievable on smaller telescopes. Raven is capable of delivering high ensquared energy into the IRCS slit. The combined technical and scientific aspects of MOAO, which Raven will demonstrate, will excite the astronomical community and build support for future facility-class MOAO instruments with much larger multiplex advantages on either 8-m class telescopes or extremely large telescopes (ELTs).

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MOAO has the potential to deliver near diffraction-limited images to multiple, small patches spread across a large FoR. One challenge of an MOAO system is that it is highly distributed. On Raven, light from one LGS and three NGSs will be sensed within a 3.5 arcmin FoR. Pixels from the OLWFS detectors will be read by the Raven Real Time Computer (RTC) and measured slopes will be used to create a tomographic model of the atmosphere above the observatory. This tomographic model will be sampled in directions defined by the position of the science probes in the patrol field and DM commands will be generated. All of these actions are performed using open-loop control. Accurate knowledge of the science probe placement in the focal plane and the relative alignment of the DMs and WFSs in the pupil plane is required. The RTC also performs the computation to provide the DM commands for the wavefront correction. Other options can be added to it. For example, the wavefront DM fitting can be added.

As a precursor to designing Raven, a broad swath of parameter space was explored in detailed end-to-end simulations [18]. This was necessary in order to determine a system architecture that can realistically meet the proposed performance require-ments and deliver useful MOAO-corrected images to the Subaru IRCS spectrograph. As Raven was conceived to be a science-capable, NGS only MOAO system, in addi-tion to a technical demonstraaddi-tion, the AO architecture was designed such that it will deliver the desired performance even when faint guide stars are used. The addition of the single, on-axis Subaru LGS to the NGS constellation improves performance and sky coverage, but does not eliminate the need for good performance with faint NGSs. The system modeling revealed that tomographic errors are the dominant factor limit-ing the performance of Raven. As a result, the performance will be highly dependent on the total amount of turbulence (and the distribution of turbulence as a function of altitude), and on the asterism of NGSs used to sense the turbulence.

Raven is capable of several operational modes that allow direct comparison of various OL wavefront reconstruction techniques:

• Closed Loop Mode: Raven will operate in classical closed-loop AO mode with bright science targets and employing the two CLWFSs.

• Open Loop (MOAO) Mode: The OLWFSs feed wavefront slope data to the RTC. The RTC performs a tomographic reconstruction utilizing OLWFS probes and science pick-off location.

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at a much slower framerate, so that the low-frequency turbulence is corrected in closed-loop directly on the science target. The high-frequency turbulence is corrected in open-loop mode.

• MOAO and Figure WFS Mode: The same basic MOAO processes happen with the Figure WFS option as well. In addition, the DM illumination source is turned on and the CLWFSs measure the spots from this source (it is assumed that the science target in this case is much fainter than the DM illumination source). The CLWFSs are read out by the RTC. The slopes from the Figure WFSs can conceivably be used in a CL mode to ensure the DM takes on the shape that is commanded by the OLWFSs.

Figure 1.3: Functional optical block diagram of RAVEN. Dashed blocks are deployable. Raven consists of 8 main subsystems: the deployable Calibration Unit, the Open-Loop NGSWFSs, the Science Pick-offs, the Science Relays, the Closed-Loop NGS Truth/Figure WFSs, the Beam Combiner, the LGSWFS and the Acquisition Camera.

1.3 System Calibration Challenges Posed by an

MOAO Instrument

To calibrate any AO instrument, misregistrations between the WFS and the DM must be measured and corrected [19]. However, for a MOAO system, the OLWFSs do not

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Figure 1.4: Top view of the Raven computer-aided design.

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see the DM so the classic approach of measuring DM-to-WFS interaction matrices cannot be used. To overcome this issue, Raven employs a CLWFS and a common DM (calibration DM in the CU). With these components, one can measure the open-loop interaction matrix between the science DM and the open-open-loop wavefront sensors. In order for the RTC to generate the optimal DM shape, the AO system must be carefully calibrated. [20].

Science observations could be up to 8 hours in a single night, during which the natural guide stars and science targets are tracked by their respective pick-off arms. During tracking, the relative positions between science DMs and OLWFSs will vary and affect the interaction matrix structure. Any misregistration between the WFS and science DM axes will alter the original (i.e. static) calibration of the WFS-to-DM registration. Therefore, any subsequent misregistration creates an error in the DM-to-WFS interaction, as embodied in T (the wavefront reconstructor [21]), and results in sub-optimal MOAO performance.

An incorrect DM-to-WFS registration, induced in the Raven calibration process, could introduce significant wavefront error. Misregistration errors, which usually have minor effects in closed-loop AO (slight increase in servolag because it needs more iterations to reach the best correction), lead to static residual wavefront errors in open-loop AO systems due to a relationship between the OLWFS and science DMs registration.

The goal of the calibration is to measure the system’s misalignments and aber-rations in order to find the best transformation matrix R given in equation (1.1) [20].

−

→_{u = CT R(−}→_{s − −}→_s

0) + −→u0 (1.1)

Where −→u represents the correct voltages sent to each science DM to obtain a flat wavefront, C is the command matrix, T is the tomographic reconstructor, R is the matrix expressing the OLWFS slopes into a common space, −→s is the slopes of the 3 wavefront sensors, −→s0 is the slopes offsets and −→u0 the command offsets. In this thesis

we do not concern ourselves with the tomographic process. We instead focus on how to best determine the open-loop command (C) and transformation matrices (R).

The algorithm proposed herein, for achieving optimal registration, utilizes an ex-perimental measurement of interaction matrices between a DM and a WFS (section 1.3.2). In section 1.3.1, a brief overview of other system aberrations, that must be corrected during calibration, is also presented.

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1.3.1 Non-Common Path and Field Dependent Aberrations

Non-common path aberrations (NCPAs) result from the static optical aberrations and optical component spatial misalignments present in the optical path difference between the science output path to IRCS and the CLWFS path (blue and checkered, see figure 1.3). NCPAs are calibrated and removed prior to acquisition.

These aberrations are field-dependent (blue) because of the pick-offs’ positions in the telescope field of view. Each WFS pick-off has an offset induced by the telescope optical aberration (see figure 1.6). The dynamic nature of an MOAO system dictates that periodic corrections for these field dependent aberrations must be applied during acquisition. Prior to acquisition, a map of aberration measurements across the field is created. The aberration map supplies offset WFS values that null the measurement with respect to the center of the FOV.

Focal plane

Pick-off

RMS zernike coefficients

Figure 1.6: Field dependent aberrations in the field of view of a telescope. When changing the pick-off position in the focal plane (left), the field dependent aberration seen depends on that position (right).

1.3.2 Determination of Interaction Matrices

In an MOAO system, there is no direct interaction matrix between the OLWFS and science DM. However, in keeping with the general AO terminology, the expression will be retained herein.

A key CU component is the calibration deformable mirror (CDM). This 277-actuator DM optically links the OLWFS to the science DMs; there is no physical means to directly register the OLWFS with the science DMs.

The indirect process to find the interaction matrix for each science DM is broken down as follows (see figure 1.7):

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Figure 1.7: Simplified diagram illustrating how the interaction matrices are record for the calibration. The light beam is generated in the calibration unit by the natural guide star source. It is bounces off the calibration DM (CDM). Then 2 pick-offs send the light to the open-loop wavefront sensor (OLWFS) and the science path with a science DM and a closed-loop WFS.

(i) Science DM-to-CLWFS interaction matrix

sβ = MβCL∗ uβ (1.2)

Where:

sβ: slopes of the CLWFS,

M_βCL: interaction matrix between the science DM and CLWFS, uβ: science DM commands.

(ii) CDM-to-CLWFS interaction matrix

sβ = MβOL∗ uc (1.3)

Where:

sβ: slopes of the CLWFS,

M_βOL: interaction matrix between the CLWFS and the CDM, uc: CDM command voltages.

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(iii) CDM-to-OLWFS interaction matrix

sα = MαOL∗ uc (1.4)

Where:

sα: slopes of the OLWFS,

M_αOL: interaction matrix between the OLWFS and the CDM.

Employing the equations above, the relationship between uβ and sα is given by:

sα = MαOL∗ (MβOL)

†_{∗ M}CL

β ∗ uβ (1.5)

Where, the double product MOL

α ∗ (MβOL)

†_{∗ M}CL

β is the SDM-to-OLWFS interaction

matrix.

It can be noted that the transformation matrix R can be expressed using these interaction matrices :

R†= M_αOL∗ (M_βOL)† (1.6)

An interaction matrix is typically rectangular (its dimension is the number of DM actuators (rows) by the number of valid lenslets of the WFS (columns)) and the pseudo-inverse with singular value decomposition is employed to invert it. In figure 1.8, the circle represents the zone seen by the subapertures of a WFS and the dots represent the actuators.

One issue in this particular case is the position of the DM edge actuators, especially ones outside the pupil. Their effect is not well-measured because their centers are located outside of the zone seen by the WFS. If this effect is not taken into account, these actuators will tend to be commanded by higher voltages because the WFS barely measures the effect of an edge command. This can also result in damaging the DM. A solution to this problem is to set the singular values associated to deformations that cannot be seen by the WFS to zero. The edge actuators are then not used to control high spectral frequencies. This reduces the noise propagated by the command matrix.

We can avoid performing a pseudo-inversion by developing a theoretical and ex-perimental technique to relate the spatial position of the system’s DMs actuators with the WFSs subapertures.

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1.4 Previous Work

1.4.1 Registration Parameters

In closed-loop AO systems, DM-to-WFS misalignments are accounted for in the inter-action matrix created during calibration. Thus, the closed-loop control will flatten the wavefront even if the command is not perfect. In open-loop, however, the misalign-ments must be identified and quantified prior to their integration in the command matrix because the DM must accurately go to the correct shape with no feedback. There is no published method for applying the interaction matrix in order to locate DM actuator positions on the WFS camera and, subsequently, using these locations to extract misregistration parameters (magnification, translation and rotation). The misregistrations created by these misalignements are particularly important for open-loop tomographic wavefront reconstruction. More transformations could be studied such as barrel deformation but the method will show that the five presented geomet-rical transformation can provide a good model.

Neichel’s paper [22] presents a method to identify the misalignment parame-ters from an interaction matrix for alignment and tomographic reconstruction. The method is based on registration of misalignment in interaction matrices. It relies on an iterative method of identification of the parameters to relate an estimated matrix to an experimental one.

An alignment technique for a DM and a WFS, presented by Oliker [23], measures misalignments, such as horizontal and vertical translation, magnification and rotation, using a DM with its actuators positioned in a waffle mode. Significant advantages are realized by employing the actuator mapping and misregistration technique:

• The precise spatial location of each DM actuator, as projected onto the OLWFS or CLWFS, is generated.

• A subset of DM actuators can be used to accurately determine the misregistra-tion.

• The method utilizes an interaction matrix, which is always experimentally de-termined during Raven calibration.

This thesis will use the same algorithm for the horizontal and vertical translation. However, the rotation and magnification misregistration parameters will be estimated by developing other geometrical methods.

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1.4.2 Interaction and Transformation Matrices

The transformation matrix presented in equation 1.1 can be expressed as the product of two interaction matrices (see eq. 1.6). These are usually the experimental inter-action matrices measured on the instrument. In this thesis, I am proposing a new method to create this transformation matrix synthetically using the instrument regis-tration parameters. This will provide a noise-less theoretical transformation matrix.

Another way to find this transformation matrix has been developed for the CA-NARY instrument [24] using the Learn method from their Learn and Apply algorithm [25]. The Learn method creates the transformation matrix experimentally without expressing the registration parameters. It links the WFSs data by measuring the covariance matrices between the WFSs. This Learn algorithm is a very powerful tool because all the misalignments are then considered, even the higher orders of parame-ters that I do not consider. The problem also lies in this advantage that the method proposes: there is no knowledge of the instrument’s registration parameters between the optical components. If one wants to model the system, these parameters cannot be taken into account and their contribution is then hard to quantify.

1.5 Thesis Outline

This thesis is based on the following outline:

Chapter 1 presents the context of the thesis with the specific challenges faced by an MOAO system.

Chapter 2 describes the research objectives using the new method to register a DM with a WFS which leads to applications on the Raven instrument.

Chapter 3 presents the algorithms to map the positions of DM actuators on a WFS and extracting the registration parameters.

Chapter 4 gives the methodology to create theoretical interaction matrices and com-mand matrices from experimental measurements. This is a direct application of Chapter 3.

Chapter 5 includes the experiments performed on the Raven instrument. First, there is an overview of tests performed on the calibration unit to assess the performances of its most important components. Then the improvement of the

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alignment procedure will be presented in details using the methods developed in Chapter 3. Finally, the performances of synthetic interaction and command matrices compared to experimental matrices will be shown.

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WFS Zone not seen by WFS

Figure 1.8: Illustration of the science deformable mirror actuators as seen by the OLWFS. The three WFSs are rotated with respect to the science DM by −90◦, 180◦ and 90◦

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Chapter 2 Research Objectives

The research objectives include the following steps:

• Analyzing the problems currently preventing the optimal calibration of the open-loop AO system.

• Proposing a new technique for the calibration employing a novel registration method between critical AO components, i.e. the system’s science and calibra-tion DMs and the open-loop wavefront sensors.

• Developing a practical Raven system optical calibration tool applicable to any DM and WFS alignment operation : e.g. laboratory or prior to on-sky mea-surements.

2.1 Actuator Mapping and Misregistration

Char-acterization

There are several applications made possible by our actuator mapping method. The actuator mapping method can be used during alignment of an AO system to accu-rately center the DM with the WFS or vice-versa. This method can also be used for optical tests, such as the measurement of the pupil size on the WFS (magnification error) and the image distortion. The relative position of the DM actuators to WFS subapertures will help in the calibration of MOAO systems. The usual method to compute a command matrix uses a singular value decomposition to obtain singular values of the matrix. These singular values are then thresholded experimentally to

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remove the effect of the deformations that cannot be seen by the WFS. One of the direct applications of our method for determining the relative position of the DM actuators to the WFS subapertures is the optical alignment on the bench. When the actuators’ positions are displayed in real-time on the WFS camera, the user can posi-tion these actuators to align the DM according to the design. The common alignment procedure is to superimpose the DM center actuator on the WFS center.

Another important application from the position of the DM actuators relative to a WFS is the characterization of the registration. The registration parameters are based on the actuator mapping and give quantitative data for the alignment. A noise-less theoretical transformation matrix can be built from the registration parameters. Even though there will be noise in those registration parameters, we can measure the five required parameters more accurately and faster than we can measure an exper-imental interaction matrix. The registration parameters considered here are (i) the horizontal and vertical shift, (ii) the rotation angle and (iii) the horizontal and ver-tical magnification between two sets of actuators’ maps. These actuators’ maps can be strictly experimental, in which case the parameters will give information on the relative alignment of the WFSs and DMs on the bench. The misregistration parame-ters will also be utilised by the Raven tomographic reconstructor. That reconstructor needs the measurements and commands expressed in the same space. This will be done using the misalignments parameters between WFSs and DMs.

(a) Rotation (b) Magnification (c) Translation

Figure 2.1: Illustration of the misregistration parameters. The black grid is the reference, the red grid is the misregistered one.

We have developed a method for estimating the misregistration parameters for the Raven instrument. It is applicable to systems with membrane-style DMs and Shack-Hartmann wavefront sensors. The method relies on the fact that the mirror membrane is continuous and would not work on a segmented DM. A different number of actuators

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or a different geometric pattern would work as long as the mirror has a continuous membrane. This method is applicable to Shack-Hartmann wavefront sensors because the positions of the actuators are determined directly by the displacement of the WFS slopes. However, the method could be extended to other types of WFSs.

2.2 Synthetic Interaction and Command Matrices

The registration parameters are the base to create theoretical rotation matrices. They are the matrices that transform the data from the OLWFS space into CLWFS space. They are built using experimental measurements. They are expected to be as good or better than the experimental matrices because they will not contain measure-ment noise except from the registration parameters identification. In chapter 4, the construction of the synthetic matrices will be presented before their performance com-parison to an existing method, called Learn and Apply, transforming WFS data from one space to another in chapter 5.

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Chapter 3 Improving the Calibration of an

MOAO Instrument

3.1 Actuator Mapping Method

To determine the position of the actuators of a deformable mirror, we can use the interaction matrix between that DM and a wavefront sensor. Although a minimum of 3 actuators are required to compute the 5 registration parameters, we use informa-tion from a larger selecinforma-tion of actuators to maximize accuracy. Interacinforma-tion matrices are used because they are always measured during the calibration process but only information from selected actuators inside the pupil are retained.

Each column of the interaction matrix shown in figure 3.1 represents the coor-dinates of the WFS slopes during an actuator poke. Since the actuators are poked sequentially, there are as many columns as actuators on the deformable mirror. The upper half of the matrix gives the X coordinates of the slopes whereas the lower half gives their Y coordinates.

Figure 3.2 shows the position of the WFS spots from the interaction matrix when actuator number 107 is poked (blue crosses). When the DM is in its flat position, the spots are located at their reference positions (red crosses). We chose to express slopes in terms of the fraction of the FOV of the WFS subapertures. They are given in pixels so there is a need to convert them into a general coordinates reference. In this case, the selected coordinate system reference is the WFS subaperture but the reference could be chosen arbitrarily because it would result in a simple shift of the coordinates.

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Figure 3.1: Experimental interaction matrix between a CLWFS and a SDM. Each column of the matrix represents the coordinates of the WFS slopes during an actuator poke.

0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 90

Slopes positions for actuator #107 poked on WFS camera

reference position slope when actuator is poked

Figure 3.2: Example of WFS reference positions (red) and slopes (blue) when actuator 107 is poked.

Actuator positions are determined from the intersection of slopes resulting from the push of a single actuator. The slopes on the WFS point towards the actuator position. The actuator, the reference position of each slope and the slopes seeing the effect of a poke form a straight line. Thus the intersection of the lines with each other

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gives the position of the actuator (blue lines in figure 3.3). This method works when only one actuator is poked, or when there is no overlap between the displacement of the spot of two actuators. In the latter case, their zones of influence should be distinct (see section 5.2). If a subaperture is affected by two actuators, it will not point at an actuator’s location and will add noise to the measurement.

Figure 3.3: Illustration of how the intersection of the slopes gives the actuator position (blue lines). For small spot displacements, the accuracy is lost and intersect the other lines far from the actuators position (red line).

In the experimental determination of the intersection point (see figure 3.3), mea-surement noise must be removed through the selection of an appropriate filter. WFS noise induces random errors on the slope measurements. In figure 3.3, the red line drawn by the slope on the bottom-left does not pass through the actuator position. This error comes from the noise of the measurement. The threshold chosen to filter the slopes is 10 percent of the maximum spot displacement. If the spot’s displacement is less than that value, it will be set as a null displacement and will not be used in the intersection method. The slope errors are not accounted for in the threshold, as a result, an optimization algorithm is performed to find the actuators’ positions.

3.1.1 Optimization Method for the Actuators Best Position

The position of each actuator is computed independently and as a result, the algo-rithm has to be applied as many times as there are actuators. As illustrated in figure 3.3, the slopes do not intersect at a single point. The following optimization algorithm is used to find the actuators’ optimal positions given the displacement of the spots.

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The optimization method requires a starting point for the search process. The initial guess is chosen with the goal of being as close as possible to the actuators coordinates. This minimizes the computation time as fewer iterations will be needed. When an actuator is poked, the subapertures closest to its position have the greatest spot displacements. If the actuator is located exactly at the center of the subaperture, the corresponding spot should not move because the average slope engendered by that actuator over the subaperture is zero. That is not a problem as surrounding slopes would still point toward the actuator position.

The goal of the algorithm is to minimize the distance between a point and the lines prolonging the slopes. In figure 3.4, the arrows represent the spot displacement from the center of the subaperture. A zone is defined by two dotted lines which have the following parameters: origins are the center of the subapertures and directions are the slope plus or minus ten degrees. This define an arbitrary error zone related to the slope error. Then for the line closest to the considered point, r and σ are defined as: r is the distance between the estimated point and its projection B onto the slope, and σ is the distance between A and B. A is the projection of the guessed point onto the closest error line.

For each non-thresholded slope, r and σ are computed and the function to optimize is equation (3.1) where d is the slope length and n the number of non-thresholded slopes. Multiplying by d2 _{gives more weight to spots with longer displacement and}

reduces the noise. Matlab has the fminsearch function that requires a function to minimize and a starting point. With the initial guess and equation (3.1) to minimize, the actuators positions are obtained : they are optimized to be as close as possible to every slopes considered. The optimization process will find the point that is closest to each prolonged slopes and inside the error cone delimited by the error bars. In a case of only 2 slopes measured, the optimized point will be the intersection of the prolonged slopes. = n X k=1 d2_kr 2 k σ2 k (3.1)

3.1.2 Disk Filtering : Actuators Visible on WFS

The optimization procedure generates a map of actuator positions. Figure 3.5a presents these positions in red dots and the blue crosses represents the center of the WFS subaperture. Most of the actuators outside of the camera have a position

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that is harder to find. This can come from the fact that there are not enough data from the slopes and also because the edge of the mirror membrane is stiffer (so it moves less). The influence functions of the edge actuators are different than other actuators. If the goal is to filter the actuators located outside of the WFS camera, the information in figure 3.5a is enough. The user just has to locate the pupil on the WFS and relate it to the position of the blue crosses.

However, characterising the misregistration parameters requires a more accurate map so the edge actuators are removed automatically through an algorithm that selects an arbitrary zone. That zone is generated via the subapertures of the WFS and keeps all the actuators located within 4.5 subapertures from the center of the WFS. Considering the number of registration modes, not all actuators are needed (only 3 actuators would be enough in theory).

The robustness of this method for measuring WFS to DM registration allows the user to apply it to DMs with different number of actuators and geometries. Figure 3.6 presents the maps of two DMs on the same WFS. On the left (figure 3.6a) is the map for the science DM which has 145 actuators, whereas on the right side (figure 3.6b), the CDM has 277 actuators. The latter has more actuators on the same pupil so they look closer on the WFS. It has to be noted that because of the disk filtering part of the algorithm, the number of actuators on the maps are lower than the total number of actuators.

3.2 Extracting Misregistration Parameters from Maps

of Actuators

Once we have measured the relative locations of DM actuators and WFS subaper-tures, we can extract the misregistration parameters. We compare the actuator map measured by the WFS and the expected manufacturer specified actuator map (fig-ure 3.7). Two actuator maps to compare : a reference map (red) and a model with misregistration (blue).

The transformation from a point A on the misregistered map to a point Aref on

the reference map (the DM frame) is given by:

A = cos θ sin θ − sin θ cos θ ! ∗ xM ag 0 0 yM ag ! ∗ Aref + xShif t yShif t !! (3.2)

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Where: xM ag and yM ag are the horizontal and vertical magnifications, θ is the rotation

angle between the maps, the origin of the rotation is the center of the DM and xShif t

and yShif t, the respective horizontal and vertical shifts. Identifying these parameters

is the goal of this section. Using equation (3.2), the rotation angle and magnifications are identified first [26].

The rotation angle is computed using the dot product and cross product of all the possible combinations of vectors inside each map. Noting −→u one possible vector in the reference set and −→v the corresponding vector in the measured map, the dot product is given in (3.3) and the signed norm of the cross product is given in equation (3.4).

−

→_{u .−}→_{v = kuk.kvk. cos(\}−→_{u , −}→_{v )} _(3.3)

k−→u × −→v k = kuk.kvk. sin(\−→u , −→v ) (3.4) Dividing the two previous equations and taking the resulting arctangent, gives the signed rotation angle (equation (3.5)).

θ = arctan k−

→_{u × −}→_{v k} −

→_{u .−}→_v (3.5)

Since a rotation preserves the relative distances inside a rotated set, there is no need to apply the inverse rotation to the points before extracting the magnification. This is possible when taking the DM as reference but also because the actuators’ order has been recorded during the DM mapping. The DM frame is compared to a theoretical reference DM frame so if there is a magnification in any other direction that is not X nor Y, it will result in a projection on the X and Y axes that will be negligible to X and Y magnification. There is a difference in the direction of the magnification but the method to compute the magnification in each direction is identical:

• xdistancesand xdistancesREF the distances of all the possible combinations between

two actuators lined up horizontally, for the misregistered map and the reference one, respectively (see figure 3.8a),

• ydistances and ydistancesREF the distances of all the possible combinations between

two actuators lined up vertically, for the misregistered map and the reference one, respectively (see figure 3.8b).

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Applying the formulae in equations 3.6 and 3.7 gives the horizontal and vertical magnifications, respectively. They are the root mean square of the sum of all the squared distances considered in the misregistered map divided by the sum of all the squared distances considered in the reference map.

xM ag = s Pn k=1x2distances(k) Pn k=1x 2 distancesREF(k) (3.6) yM ag = s Pn k=1ydistances2 (k) Pn k=1ydistances2 REF(k) (3.7) Accordingly to equation (3.2), the data (see figure 3.9a) is unrotated and unscaled. The rotation is performed using the center actuator as the pivot point (see figure 3.9b). The shift is the displacement of the center of gravity of the grid. When it is removed, the experimental map (blue) can be superimposed to the reference map (in red, see figure 3.9c).

These registration parameters are included inside the getRegistrationParameters function. It takes an interaction matrix as input and outputs the registration param-eters in a registrationParamparam-eters structure. The Matlab code for the function and the structure are presented in Appendix A.2 and A.1.

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Figur e 3.4: Il lustr ation of the optimization function for two slop es. The cr oss repr esents the p oint to optim ize.

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−40 −20 0 20 40 60 80 100 120 140 −20 0 20 40 60 80 100 120 Actuator Map Center Of Subaperture

(a) Map of actuators (red dots) compared to the positions of the subaper-ture centres (blue crosses)

−40 −20 0 20 40 60 80 100 120 140 −20 0 20 40 60 80 100 120 Actuator Map Center Of Subaperture

(b) Same as 3.5a but with a filtering disk: actuators positions found out-side the circle are discarded in the mapping process.

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0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Actuator Map of SDM on CLWFS

(a) Map of SDM actuators

0 20 40 60 80 100 10 20 30 40 50 60 70 80 90 Actuator Map of CDM on CLWFS (b) Map of CDM actuators

Figure 3.6: The actuator mapping is applied to DM with a different number of actuators: 145 for SDM and 277 for CDM.

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0 20 40 60 80 100 10 20 30 40 50 60 70 80 90

Example of Matrices to analyse _{Reference Actuator Map} Distorted Actuator Map

Figure 3.7: Comparison of 2 actuator maps. The red one is the reference. The blue one is rotated by an angle of 40◦ and shifted by 5 percent of a subaperture on the WFS. The blue lines show the corresponding actuators.

(a) Illustration of the combination of distances for the horizontal magnifi-cation

(b) Illustration of the combination of distances for the vertical magnifica-tion

Figure 3.8: Illustration of the combination of distances. The blue dots represent the actua-tors positions and the arrows represent the distances between all the possible combinations of these points.

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−5 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4

Example of Matrices to analyse

Reference Actuator Map Experimental Actuator Map

(a) Experimental map extracted from a SDM-to-CLWFS interaction matrix (blue crosses) to compare to a reference map (red crosses). The lines indicate the cor-responding actuators in each map.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4

Rotation and Scale removed

(b) Rotation, scale removed from the data to fit to the reference actuator map before extracting the translation parameter.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4

Rotation, Scale and Shift removed

(c) Rotation, scale and translation removed from the data to fit to the reference actuator map.

Figure 3.9: Experimental results of the extraction of the registration parameters from a SDM-to-CLWFS interaction matrix.

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Chapter 4 Creating Command Matrices with

Synthetic Influence Functions

In this chapter, we consider two ways to create synthetic command matrices. The first one uses the interaction matrices described in sections 4.1 and 4.2. The second method, described in section 4.3, is based on the fitting of the DM phase.

4.1 Creating Synthetic Interaction Matrices

Synthetic interaction matrices are created from the registration parameters of ex-perimental matrices and the OOMAO tool. The Object-Oriented Matlab Adaptive Optics (OOMAO [27]) is an extension of the Matlab language. Its library contains a set of classes developed to perform numerical modelling of an AO system and allows the user to propagate a wavefront through the system. This is one of the modelling tools used in the Raven project. The OOMAO simulation tool was not adapted for the generation of synthetic interaction matrices. It had the option to add some misalignments between a DM and a WFS but this was not accurate enough for our purpose. The problem lies in the definition of the phase propagated to the WFS. This is defined by the finite number of pixels on the WFS camera. For Raven modelled with OOMAO, the phase is a matrix of 120x120. Then, when misregistrations are added to the phase, there is a loss of information due to sampling. For example, if a phase needs to be shifted on a vertical or horizontal axis by a fraction of a pixel, interpolations of the new values in the matrix have to be performed. That interpola-tion adds error when the transformainterpola-tion does not make a pixel correspond to another

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pixel of the phase.

Another limitation of that method is that there is no easy way to simulate a DM with actuators outside the pupil. All the DM actuators are defined on the pupil. As a result, for our case of science DMs with 13x13 actuators (with only 11x11 on the pupil), the user has to remember to use the OOMAO model with a 11x11-actuator DM. This does not directly work when the matrices are to be used with the laboratory instrument because of the different size of the matrices.

A different approach is to use OOMAO and simulate the DM poke to the WFS without a DM. The goal is to present the WFS with a series of poked DM actuators in the form of influence functions placed where they would have been if the DM was poked. These influence functions are presented as phase maps. Because OOMAO relies on the phase propagation to compute slopes, this is a realistic method to create a synthetic interaction matrix.

The first step is to compute the DM influence functions as close to reality as possible. To perform this task, we measure the influence functions using a Zygo interferometer. This measurement confirms the manufacturer data: 30% coupling for the CDM and 40% coupling for the SDM. The influence functions also give information on the amplitude corresponding to a normalized manufacturer DM command poke. Then, with that data, an analytic expression of the influence function is computed using a Gaussian function. This function is chosen because it is easy to compute for these tests and should validate the method. This Gaussian then has to be centered on the poked actuator’s position as seen by the WFS. I developed a Matlab function, getSyntheticMatrix, to compute a synthetic matrix directly from an experimental one (Appendix A.3).

The getSyntheticMatrix function use all experimental interaction matrices gener-ated on the Raven instrument and creates synthetic matrices. The size of the inter-action matrices is dictated by the DM which is simulated. getSyntheticMatrix also uses the influence functions and the number of actuators with their configuration on the mirror. These matrices have the following dimensions: 160x277 for the CDM or 160x145 for the SDM. With this knowledge, we measure the registration parameters to know how the DM is oriented and scaled with respect to the WFS.

The map of valid actuators is created from the DM parameters. The SDM and CDM do not follow the same actuators’ placement, nor the same number of actuators. After the grid is created, it has to be scaled, rotated and shifted to appear as the real DM seen on the WFS. The measurements confirm that the CDM has a 30%

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coupling and the SDMs have a 40% coupling (figure 4.2). Although the scales from the Zygo interferometer and the synthetic WFS phase are different, due to the higher resolution of the Zygo camera, we can compare the influence functions obtained by both methods.

Figure 4.1: Wavefront measurement setup with a Zygo interferometer (left) and a DM (right).

A Gaussian is a simple way to describe an influence function and is fit to validate the method. The coupling and amplitude are taken from the measurements done with the Zygo interferometer. The coupling will affect the width at half maximum. The Gaussian function is defined as follows (eq. 4.1):

gaussian(x, y) = Ae−(

(x−x0)2

2σ2_x +

(y−y0)2

2σ2_y ) _(4.1)

With A the amplitude, x0 and y0 the coordinates of the center of the peak, and σx

and σy the spreads of the function. The spread of the function is the same in both

directions because the membrane is isotropic, except near the edges. It is set to have the right coupling for either SDM or CDM. The coordinates of the center of the peak are the coordinates of the actuator considered while registering the interaction matrix. We manage to match the coupling but the Gaussian appears to have a broader peak and more narrow wings than the actual influence function. As seen on the graphs, the first intersection with the adjacent influence function for the experimental measurement is 64% while it is 73% for the Gaussian. A different function could be used for a better fit but the Gaussian function is easier to implement, debug and test. The influence functions’ amplitudes are computed so they correspond to the actual measured amplitudes. This is done separately for each actuator because the maximum amplitude is not the same, depending on the poked actuator. Edge actuators, for example, have a lower amplitude than other actuators.

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x (pixels) y (pixels) 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 −6 −5 −4 −3 −2 −1 0

(a) CDM phase on Zygo

(b) ZYGO CDM influence functions (actuators 135 to 143) Figure 4.2: (a) Zygo CDM phase and (b) influence functions.

Once the influence functions and the grid of actuators are created, the complete interaction matrix can be modelled. The method presented here uses a basic series of successive actuator pokes while recording the WFS slopes. An interaction matrix

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x (pixels) y (pixels) 20 40 60 80 100 120 20 40 60 80 100 120 −6 −5 −4 −3 −2 −1 0

(a) Synthetic CDM Phase generated with Gaussian

(b) Synthetic CDM influence functions (actuators 138 to 145) Figure 4.3: (a) Synthetic CDM phase and (b) influence functions.

registers the effect of DM commands on a WFS. In this case, the DM is simulated by a series of phase screens propagated from the OOMAO source object to the WFS. There are as many phase screens to present to the WFS as there are DM actuators.

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Every point on the synthetic grid has to correspond to an actuator on the physical DM and must be correctly oriented and positioned with respect to the WFS. For each phase map presented and propagated to the WFS, the slopes are saved. Slopes measured from successive actuator pushes build interaction matrix column by column. The creation of synthetic interaction matrices is done with a function that takes the experimental interaction matrix as input and outputs the synthetic matrix. The process gives a synthetic matrix that is qualitatively the same as the experimental one. We still need to validate our results experimentally to show their performance during the correction. Figure 4.4 shows that the resulting interaction matrix is similar to the experimental one. As a result, the next step is to test them in an AO setup.

4.2 Command Matrix Using Singular Value

De-composition

The singular value decomposition (SVD) is a classic method to compute the command matrix using the interaction matrix between the SDM and CLWFS. An interaction matrix presents the conversion matrix between an action on the DM as seen on the WFS. In order to find how to control the DM given the data collected via the WFS, a direct approach is to invert the interaction matrix. However, since the number of DM actuators does not equal the number of WFS subapertures, the interaction matrix is rectangular and a pseudo-inversion is required. This decomposition is performed in Matlab with the svd() function. Then singular values are extracted and filtered with a threshold. These singular values are filtered out for better performances. The following code shows how the command matrices are computed :

1 [U,S,V] = svd(interactionMatrix); % Singular Value Decomposition 2 singularValues = diag(S); % Collect the singular values

3 nThresholded = 53; % Number of modes to threshhold

4 iS = diag(1./singularValues(1:end−nThresholded)); % Inversion of ...

matrix with thresholded singular values

5 [nS,nC] = size(interactionMatrix);

6 iS(nC,nS) = 0; % Fill the matrix with zeros to have the correct ...

dimensions

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Figure 4.5a gives an example of the inversion of the SDM1-to-CLWFS1 interaction matrix. The result is the command matrix (figure 4.5b). The same can be obtained with synthetic matrices (figure 4.6) and the performances of synthetic command ma-trices are presented in chapter 5.

4.3 Command Matrix Using Phase Fitting

A command matrix generated to fit the distorted phase directly, using the actuators’ influence functions, has the advantage of avoiding the singular value decomposition of an interaction matrix.

The creation of the command matrix using a phase fitting method to correct the distorted wavefront is done with the getCommandMat function that I developed in Matlab. The phase to correct is expressed into the CLWFS’s space. It takes a SDM-to-CLWFS interaction matrix as input and it outputs the command matrix directly as well as the matrix of influence functions that can be used directly in Matlab for phase fitting using the mldivide operator on the phase in Matlab. However, the function has to produce the command matrix because the correction process of the Raven sequencer does not handle the mldivide Matlab operation needed to fit the phase (the RTC only handles multiplication between matrices). A pseudo-inverse of the influence functions’ matrix is then required to perform the phase fitting.

To create the command matrix, a synthetic interaction matrix has to be created first. It follows a similar method as that described in section 4.1 with a few differences. The getRegistrationParameters function is used to extract the registration parameters between the SDM and CLWFS. It must be noted that the CLWFSs for each science path are used as references (all the WFS slopes are expressed in CLWFS space). The registration parameters position the influence functions as they would appear on the CLWFS. The set of influence functions will recreate the fitted phase to apply the correction.

The influence functions are also created with a Gaussian function that respects the actuator coupling of the DMs and the amplitude. Again, the influence functions are centered on the actuators. Then the influence functions for each actuators are stored in a matrix where each column of the matrix represents the expression of the actuators’ influence functions from 1 to 145 (there are 145 actuators on the SDM). A pseudo-inversion of this matrix will give the phase fitting command matrix. An example of the phase fitting is given in figure 4.7. On the left, there is the

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non-corrected phase simulated via OOMAO. This phase is fitted with the DM using a synthetic command matrix (middle) and the residue is presented on the right. This method has not been tested at the time of writing this thesis because the data would need to be in the phase space rather than in the current slope space.

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actuator # slopes 20 40 60 80 100 120 140 20 40 60 80 100 120 140 160 −4 −3 −2 −1 0 1 2 3 4 5

(a) Experimental Interaction Matrix SDM2toCLWFS2.

actuator # slopes 20 40 60 80 100 120 140 20 40 60 80 100 120 140 160 −5 −4 −3 −2 −1 0 1 2 3 4

(b) Synthetic Interaction Matrix SDM2toCLWFS2.

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Actuators Slopes 20 40 60 80 100 120 140 20 40 60 80 100 120 140 160 −3 −2 −1 0 1 2 3

(a) Experimental interaction matrix SDM1-CLWFS1

Slopes Actuators 20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2

(b) Experimental command matrix obtained after SVD.

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Actuators Slopes 20 40 60 80 100 120 140 20 40 60 80 100 120 140 160 −3 −2 −1 0 1 2 3

(a) Synthetic interaction matrix SDM1-CLWFS1

Slopes Actuators 20 40 60 80 100 120 140 160 20 40 60 80 100 120 140 −0.15 −0.1 −0.05 0 0.05 0.1 0.15

(b) Synthetic command matrix obtained after SVD.

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(a) Aberrated phase simulated on OOMAO.

(b) Phase fitted with the synthetic command matrix.

(c) Residue after correction. Figure 4.7: Phase fitting.

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Chapter 5 Experiments

5.1 Calibration Unit Performance Tests

5.1.1 Calibration Deformable Mirror tests

Before aligning and validating the performance of the Raven AO system, the calibra-tion unit (CU) was studied to expand the knowledge of the CU behaviour beyond the expected requirements [28]. Here, we describe tests of the calibration DM and the source simulating the NGS.

The setup of the tests was as follows: the Zygo interferometer was positioned in front of the DM that was fixed to the table. The five cables sending commands to the DM were attached to a post so they would not pull on its back and change its position during testing. The Zygo interferometer can be controlled remotely through the network (Figure 5.1). Since the Zygo computer cannot be used for anything besides controlling the interferometer and getting data from it, a second computer (Master-PC) is connected to the DM and can access the Zygo-PC through the network. The setup is controlled under Matlab from the Master-PC dedicated to do the tests, collect and analyse data.

Thermal Stability

At the start of the tests, the DM was initialized to a DM best flat position, then measurements were taken throughout a night and the DM shape was analysed. From the beginning to the end of the test, the DM did not receive any new command. After launching the test, there was no one in the room to perturb the measurements during

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Zygo

DM

Zygo

PC

Master PC (Matlab)

Network

Figure 5.1: Set-up of the Zygo interferometer in front of the DM.

16 hours, so perturbations only came from changes in temperature (see figure 5.2). The tip and tilt (blue line) were caused by the misalignment between the DM and the Zygo interferometer. We removed it to isolate the DM stability measurement from the mechanical alignment of the DM with the interferometer (red line). There is a drastic increase of the wavefront error during the first 20 minutes. Then this wavefront residue stays stable during 15 hours. We see that there is a warm-up period of about 20 minutes. That means that the DM needs time to reach an equilibrium. The 10 nm RMS found on the stability of the system is very good and meets manufacturer’s datasheet.

Repeatability Tests

Repeatability tests reflect how closely the DM shape remains the same after receiving repeated commands during a night. These repeatability measurements alternate 11 phase screens (with 1 best flat) in a continuous loop (figure 5.3). There were 10 measurements per hour (every 6 minutes). When no measurements were taken, the DM is changing shape without using the Zygo to get data. That makes the DM stay warm and it simulates the DM normal use when different commands are applied throughout a night of observations to correct atmosphere turbulence. The tip/tilt data includes information about the misalignment between the DM and the Zygo interferometer. Since we are not interested in that information, we removed the tip/tilt information from the data. The repeatability was less than 10 nm RMS after

Optimizing the Optical Calibration Performance of a Multi-Object Adaptive Optics Instrument

Contents

List of Tables

List of Figures

Introduction

1.1

Multi-Object Adaptive Optics (MOAO)

1.2

The Raven Multi-Object Adaptive Optics

Demon-strator

1.3

System Calibration Challenges Posed by an

MOAO Instrument

1.3.1

Non-Common Path and Field Dependent Aberrations

1.3.2

Determination of Interaction Matrices

1.4

Previous Work

1.4.1

Registration Parameters

1.4.2

Interaction and Transformation Matrices

1.5

Thesis Outline

Chapter 2

Research Objectives

2.1

Actuator Mapping and Misregistration

Char-acterization

2.2

Synthetic Interaction and Command Matrices

Chapter 3

Improving the Calibration of an

MOAO Instrument

3.1

Actuator Mapping Method

3.1.1

Optimization Method for the Actuators Best Position

3.1.2

Disk Filtering : Actuators Visible on WFS

3.2

Extracting Misregistration Parameters from Maps

of Actuators

Chapter 4

Creating Command Matrices with

Synthetic Influence Functions

4.1

Creating Synthetic Interaction Matrices

4.2

Command Matrix Using Singular Value

De-composition

4.3

Command Matrix Using Phase Fitting

Chapter 5

Experiments

5.1

Calibration Unit Performance Tests

5.1.1

Calibration Deformable Mirror tests

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